Design of phononic crystal using open resonators as harmful gases sensor

This paper investigates the ability to use a finite one-dimensional phononic crystal composed of branched open resonators with a horizontal defect to detect the concentration of harmful gases such as CO2. This research investigates the impact of periodic open resonators, defect duct at the center of the structure, and geometrical parameters such as cross-sections and length of the primary waveguide and resonators on the model's performance. As far as we know, this research is unique in the sensing field. Furthermore, these simulations show that the investigated finite one-dimensional phononic crystal composed of branched open resonators with a horizontal defect is a promising sensor.


Sensor configuration and equations
In Fig. 1, a schematic of the 1D-PnC composed of branched open resonators is proposed. The main guide has a cross-section S 1 and a thickness d 1 . The branched open resonators have cross-section S 2 and height d 2 . The proposed 1D-PnC comprises branched-open resonators sensor, and a defect guide sandwiched between two PnCs. The structure will be filled with gas samples containing different concentrations of CO 2 . The plane wave theory can be used for stationary samples inside the sensor, and the effects of temperature gradients, higher-order modes, and viscosity effects are neglected 23 .
The theoretical method used to study the response of the proposed periodic branched open resonators to the incident acoustic waves is called the transfer matrix method (TMM) as the following 23-30 : where A i = cos k d i 2 , B i = jZ i sin k d i 2 , C i = j Z i sin k d i 2 , D i = A i , k = ω / c is the wave number, ρ is the density, Z i = ρc S i is the impedance of each period of the proposed branched open resonators, and c is the acoustic speed. The acoustic pressure at the end of the opened lateral chimney is approximately zero, and the acoustic admittance of the acoustic wave ( y R ) is calculated as: For the defect cell:

Results and discussions
As an initial condition, the geometrical parameters of the main guide and open resonators of the proposed sensors will be N = 10, d 1 = 0.6 m, d 2 = 0.15 m, d d = 0.3 m, S 1 = 1 m 2 , S 2 = 0.75 m 2 , and S d = S 1 m 2 . Table 1 shows the acoustic properties of an air sample at different concentrations of CO 2 . The gradient of the density of the sample from low to high and acoustic speed from high to low with the increase of the CO 2 concentration ensures that both density and acoustic speed can be considered an indicator of the concentration of CO 2 .
The transmittance (red spectra) and dispersion relation (blue spectra) curves versus frequency of the proposed 1D-PnC composed of branched open resonators without defect are plotted and coincided using TMM and Bloch's theorem in Fig. 2A. In the frequency range of concern, two PhBGs extend from 1429.2 to 1478.1 Hz and from 1950.6 to 2000.6 Hz. The proposed 1D-PnC sensor composed of branched open resonators has the ability to make the PnBG due to the periodic change in the impedance and admittance of propagated acoustic waves inside the structure. By adding a horizontal defect tube sandwiched between two identical 1D-PnCs, a specific , where ∅ 1 = 1 z www.nature.com/scientificreports/ frequency of the incident acoustic wave is localized, making a defect peak inside the PnBG. This peak is very sensitive to any change in the mechanical properties of the medium inside the tubes. Considering an additional defect tube with d d = 0.3 m at the middle of the design and the other geometrical parameters having the same initial values, a resonant peak appears at the center of each PnBG, as clear in Fig. 2B. Any change in the density or acoustic speed of the gas sample due to the change in the CO 2 concentration will result in a transmittance spectrum and cause a wavelength shift to the resonant peaks and PnBGs, as clear in Fig. 3. The defect peak is redshifted to lower frequencies by increasing the concentration of CO 2 from 1975.95 Hz (at 0% of CO 2 ) to 1872.83 Hz (at 20% of CO 2 ), 1772.02 Hz (at 40% of CO 2 ), 1672.36 Hz (at 60% of CO 2 ), 1612.45 Hz (at 80% of CO 2 ), and 1575.00 Hz (at 100% of CO 2 ).
The sensitivity, figure of merit (FoM), quality factor (Q), and detection limit (LoD) of the harmful gas's sensor are used to examine the efficacy of the sensor and can be defined as follows,   where f R is the value of the resonant frequency shift with changing the acoustic speed by ( c ), and FWHM is the peak bandwidth. Sensitivity is the change in the position of the defect peak relative to the acoustic speed relative to the pure air sample as a reference. Q denotes the resonator's energy loss and is expressed as the ratio of the frequency of the defect peak to the FWHM. The sensor's ability to discover the alteration in the resonance frequency is represented by FoM 32 . LoD denotes the slightest change in the sample that can be detected. However, the FWHM gradually increases with the cross-section of S d . Besides, the T of the resonant peak records the highest intensity of (100%). Hence, the FoM and Q gradually decrease, and LoD gradually increases. Depending on the results in Fig. 5A-C, the cross-section of S d = 1 m 2 will be used in the following studies.
As d 1 increases from 0.59 m to 0.60 m, 0.61 m, and 0.63 m, the peak of the air sample is redshifted from 2001.36 Hz to 1975.95 Hz, 1950.14 Hz, and 1897.80 Hz, and the peak of the CO 2 sample is redshifted from 1595.26 Hz to 1575.00 Hz, 1554.42 Hz, and 1512.70 Hz. In Fig. 6A, the sensitivity decreases linearly with increasing d 1 . On the other hand, FWHM gradually increases with increasing d 1 . The transmittance records intensity above 99.9% for thickness d 1 higher than 0.59 m, as clear in Fig. 6B,C. Besides, FoM and Q gradually decrease, and LoD gradually increases with increasing d 1 . Therefore, a thickness of 0.59 m will be optimum. Figure 7A clears the sensitivity and FWHM versus the incident frequency for the proposed 1D-PnC sensor composed of branched open resonators with a defect cell at different values of d 2 to select the best value that gives the highest performance. The sensitivity is measured for the proposed sensor at different thicknesses of d 2 of    Fig. 9A. This redshift of the PnBG and resonant peak to lower frequencies is due to the direct proportionality between the acoustic speed of the sample and the resonant frequency according to the standing wave equation:  where d and n are the thickness and an integer, respectively. In Fig. 9B, the acoustic speed and resonant frequency versus the concentration of CO 2 are plotted. An empirical equation between the resonant frequency ( f R ) and the concentration of CO 2 ( C CO2 ) was established using the quadric fitting as the following relation: (12) f R = 0.02222C 2 CO2 − 6.392C CO2 + 2005, R 2 = 0.9975 .